### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 038, 17 pages      arXiv:2204.01445      https://doi.org/10.3842/SIGMA.2023.038
Contribution to the Special Issue on Non-Commutative Algebra, Probability and Analysis in Action

### Shifted Substitution in Non-Commutative Multivariate Power Series with a View Toward Free Probability

Kurusch Ebrahimi-Fard a, Frédéric Patras b, Nikolas Tapia c and Lorenzo Zambotti d
a) Department of Mathematical Sciences, Norwegian University of Science and Technology, NO 7491 Trondheim, Norway
b) Université Côte d'Azur, CNRS, UMR 7351, Parc Valrose, 06108 Nice Cedex 02, France
c) Weierstraß-Institut Berlin and Technische Universität Berlin, Berlin, Germany
d) LPSM, Sorbonne Université, CNRS, Université Paris Cité, 4 Place Jussieu, 75005 Paris, France

Received April 05, 2022, in final form May 29, 2023; Published online June 08, 2023

Abstract
We study a particular group law on formal power series in non-commuting variables induced by their interpretation as linear forms on a suitable graded connected word Hopf algebra. This group law is left-linear and is therefore associated to a pre-Lie structure on formal power series. We study these structures and show how they can be used to recast in a group theoretic form various identities and transformations on formal power series that have been central in the context of non-commutative probability theory, in particular in Voiculescu's theory of free probability.

Key words: non-commutative probability theory; non-commutative power series; moments and cumulants; combinatorial Hopf algebra; pre-Lie algebra.

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